Abstract: The presented work is dedicated to the robustness of a milk manufacturing workshop including time interval constraints. In such systems, operation times are included between a minimum and a maximum value. Weighted Marked Graphs are used for modelling. Some results proposing to transform Weighted Marked Graphs into Marked Graphs are reviewed, which allow obtaining a model that can be used to apply some robustness results of the state of the art. The main contribution of this paper is a computing algorithm of the maximal time disturbances allowed at a given point. Finally, to demonstrate the effectiveness and accuracy of the proposed algorithm, an application to a milk production unit is outlined. The possession of this exact value allows checking the death of marks on the levels of synchronization transitions of a P-time Petri net model without generating any false alarm. The results show that the difference between the exact value and the lower bound of the state of the art algorithm is quite important.

Many techniques for the modelling and quantitative analysis of manufacturing systems have been investigated. Among them, Coloured Petri Nets are considered as tools especially suitable for systems that exhibit concurrency, conflicts, and synchronization. Unfortunately, when there are maximum duration constraints, the modelling power of Coloured Petri Net is a disadvantage, as it becomes really difficult to prove, using properties of the net structures that the time constraints remains fulfilled.

Marked Graphs (MG) on the other hand are known for their strong analytical properties, but they do not allow the assembly or disassembly of batch components to be represented in a compact form. To take into account these kinds of processes and to reduce the size of the models, Weighted Marked Graphs (WMGs) can be used [Toursi and Sauer, 2004], [Marchetti and Munier, 2009]. WMGs are particular MGs allowing weights on the arcs. In most works, the proposed solution is to transform the WMG into an ordinary MG allowing the use of well-known methods of performances analysis [Munier, 1993], [Nakamura and Silva, 1999], [Hamaci et al., 2006]. Marked Graphs indeed are specifically designed to do performance evaluations which prove some structural properties on the graphs. However, few results based upon this tool address the time disturbance robustness problem.

The systems considered with these approaches have a robustness property which allows maintaining products quality when there are time disturbances [Jerbi et al., 09]. The robustness is defined as the ability of the system to preserve the specifications facing some expected or unexpected variations. Therefore, the robustness characterizes the capacity to deal with disturbances. Passive robustness is based upon variations included in validity time intervals. There is no control modification to preserve the required specifications.

The work presented in this paper focuses on the robustness of workshops with assembling tasks, regarding time disturbances. This paper is organised as follows. Section 2 begins with the definition of the P-time Petri Nets and WMGs models of workshops. In section 3, some results about the transformation of a WMG into an ordinary MG are described. They can be used as a first step towards the study of the robustness of the considered milk manufacturing unit.

The problem of the passive robustness of manufacturing systems is tackled in section 4. The passive robustness (local robustness) of a given path in the milk manufacturing workshop is analytically built up. Lastly, a new algorithm computing the exact passive robustness margin allowed at a given node is presented.

An illustrative example is outlined and the qualitative results are discussed. Finally, some conclusions of this work are given.

M’HALLA, A., N. Jerbi, S. Collart-Dutilleul, Craye E., M. Benrejeb, Passive Robustness in Milk Manufacturing Unit with Time Constraints, in USB key proceeding of International Conference on Management and Control of Production and Logistics MCPL 2010, Portugal.